Elsevier

Fuzzy Sets and Systems

Volume 276, 1 October 2015, Pages 43-58
Fuzzy Sets and Systems

Weakening-free, non-associative fuzzy logics: Micanorm-based logics

https://doi.org/10.1016/j.fss.2014.11.020Get rights and content

Abstract

This paper deals with standard completeness for weakening-free, non-associative, substructural fuzzy logics. First, fuzzy systems, which are based on micanorms (binary monotonic identity commutative aggregation operations on the real unit interval [0,1]), their corresponding algebraic structures, and algebraic completeness results are discussed. Next, completeness with respect to algebras whose lattice reduct is [0,1], so-called standard completeness, is established for these systems using construction in the style of Jenei–Montagna. In particular, standard completeness results for the involutive logics, which was a problem left open by Horčík in the recent Handbook of Mathematical Fuzzy Logic, are provided. Finally, we briefly consider the similarities and differences between constructions of the author and Wang's Jenei–Montagna-style.

Introduction

In general, fuzzy logics are understood as logics dealing with vagueness, and such logics with truth values in the real unit interval [0,1] are currently studied. In particular, logics that characterize classes of t-norms and uninorms have been extensively investigated in recent years; infinite-valued systems Ł (Łukasiewicz logic), G (Gödel–Dummett logic), and Π (Product logic) are the most famous examples of logics based on continuous t-norms: Hájek's BL (Basic fuzzy logic) [14] and Esteva and Godo's MTL (Monoidal t-norm logic) [7], which are the logics of continuous and left-continuous t-norms, respectively, are the most important t-norm-based logics. Since t-norms have the greatest element 1 as identity, t-norm logics prove the weakening (W) φ(ψφ). Weakening-free fuzzy logic systems have been also introduced; Metcalfe's (and Montagna's) UL (Uninorm logic) [17], [18], which is the logic of conjunctive left-continuous uninorms, is the most basic uninorm logic. Uninorm functions were introduced by Yager and Rybalov [24] as a generalization of t-norms where the identity can lie anywhere in [0,1].

The starting point for the current work is the observation that fuzzy logics are substructural logics, i.e., logics that lack various structural rules such as weakening and contraction (see [10], [18]). The Full Lambek logic FL is a prominent example of a substructural logic. This system does not assume the structural rules of exchange, weakening, and contraction, but instead stipulates associativity. Substructural logics that further eliminate associativity have been introduced. Galatos (and Ono) [9], [11], [12] introduced GL, a non-associative generalization of FL whose algebraic semantics is given by the variety of residuated lattice-ordered groupoids with unit (briefly, rlu-groupoids) in the sense of [10].

According to Cintula (and Běhounek) [1], [2], a (weakly implicative) logic L is said to be fuzzy if it is complete with respect to (w.r.t.) linearly ordered matrices (or algebras) and core fuzzy if it is complete w.r.t. standard algebras (i.e., algebras on the real unit interval [0,1]). The substructural logic systems FL and GL are not (core) fuzzy logics because they are not complete w.r.t. such algebras. Their corresponding core fuzzy systems have been recently introduced. The systems UL and SL are the weakest (core) fuzzy logics extending FL and GL, respectively, denoted instead as SL in [5]. In particular, SL has been introduced very recently by Cintula, Horčík, and Noguera as the weakest possible fuzzy logic in [3], [4], [5], [16].

Although Cintula, Horčík, and Noguera have introduced non-associative substructural (core) fuzzy logics and their corresponding completeness properties, many open problems still remain. For instance, standard completeness for weakening-free involutive substructural fuzzy logics is unresolved by [16]. Note that, before introducing uninorms, Yager introduced a generalization of uninorms, a variant of the concept of uninorm obtained by removing the associativity condition in its definition. Moreover, he [22], [23] introduced a class of monotonic identity commutative aggregation (briefly, MICA) operators and asserted that MICA operators constitute the basic operators needed for aggregation in fuzzy system modeling. Let micanorm be a binary MICA operation on [0,1]. The system SLe, the SL with exchange or commutativity, and its axiomatic extensions are logic systems for MICA operations or micanorms. However, if they are the systems for MICA operations or micanorms has not yet been determined.

In this paper, we characterize weakening-free, non-associative fuzzy logics based on micanorms. Specifically, this paper introduces SLe (denoted here as MICAL) as the micanorm-based logic that is intended to cope with the tautologies of left-continuous conjunctive micanorms and their residua; its axiomatic extensions are obtained by adding axioms such as (S-INC) φ(φ&φ) and (S-DEC) (φ&φ)φ as weakening-free, non-associative fuzzy logics.2

This paper is organized as follows. In Section 2, we recall the micanorm-based logic MICAL and its weakening-free axiomatic extensions, along with their corresponding algebraic semantics. In Section 3, we define micanorms as binary MICA operations and provide some examples. In Section 4, we establish standard completeness for MICAL and its axiomatic extensions using the Jenei–Montagna-style construction introduced in [8], [15]. This consists of showing that countable, linearly ordered algebras of a given variety can be embedded into linearly and densely ordered members of the same variety, which can in turn be embedded into algebras with lattice reduct [0,1]. In particular, we provide standard completeness results for the involutive logics, which was unresolved in [16].

Note that some Jenei–Montagna-style constructions for axiomatic extensions of UL have been provided: Yang has introduced such construction for the extensions ULWt (the UL with the t-weakening axiom ((φ&ψ)t)φ) and ULcfr (the UL with the compensation-free reinforcement axiom ((φ&ψ)(φψ))((φψ)(φ&ψ))) in [25], [26]. Wang has performed similar construction for the extensions CnUL (the UL with the n-potency axiom φnφn1) and CnIUL (the CnUL with the involution axiom ¬¬φφ) in [20], [21]. The construction introduced in Section 4 is a generalization of Yang's. We may also investigate the standard completeness results in Section 4 using Wang's approach. In Section 5, we briefly consider the similarities and differences between the constructions of the author and Wang's Jenei–Montagna-style.

For convenience, we adopt notations and terminology similar to those in [2], [3], [4], [5], [6], [14], [18], [20], [21], [25], [26], and we assume reader familiarity with them (along with results found therein).

Section snippets

The logic MICAL and its axiomatic extensions

The term micanorm-based logics refers to substructural fuzzy logic systems with micanorm-based semantics, where the (strong) conjunction and implication connectives ‘&’ and ‘→’ are interpreted as a left-continuous conjunctive micanorm and its residuum, respectively. In this framework, the weakest logic is MICAL. This logic and its axiomatic extensions (henceforce referred to as extensions) can be based on a countable propositional language with formulas Fm, built inductively as usual from a set

Micanorms and their residua

In this section, by ‘1,’ ‘0,’ ‘e,’ and ‘∂,’ we denote ,, identity t, and any f, respectively, on the real unit interval [0,1]. We define standard L-algebras and micanorms on [0,1].

Definition 11

An L-algebra is standard iff its lattice reduct is [0,1].

In standard L-algebras the groupoid operator ⁎ is a micanorm.

Definition 12

A micanorm is a function :[0,1]2[0,1] such that, for some e[0,1] and for all x,y,z[0,1]:

  • (a)

    xy=yx (commutativity),

  • (b)

    ex=x (identity), and

  • (c)

    xy implies xzyz (monotonicity).

Example 1

Let identity e exist in

Standard completeness

In this section, we provide standard completeness results for Ls using the Jenei–Montagna-style construction in [8], [15].

We first show that finite or countable, linearly ordered MICAL-algebras are embeddable into a standard algebra. (For convenience, we add the ‘less than or equal to’ relation symbol “≤” to such algebras.)

Proposition 2

For every finite or countable, linearly ordered MICAL-algebra A=(A,A,,,t,f,,,,), there is a countable linearly ordered set X, a binary operationon X, and a map h

Wang's and Yang's monoids

Wang provided standard completeness for CnUL using Jenei–Montagna-style construction in [20]. His definition of the monoid operation can be regarded as a strengthening of the monoid operations introduced in Proposition 2 and Theorem 4 in the sense that, while that monoid works for CnUL, these monoids do not (see Theorem 7 (v) below). Let us call the monoid introduced in [20] Wang's monoid and the monoids introduced in Proposition 2 and Theorem 4 Yang's monoids. Here, we briefly consider

Conclusion

In this paper, we characterized weakening-free, non-associative fuzzy logics based on micanorms. In particular, standard completeness results for the involutive logics, which was unresolved in [16], are provided.

We established standard completeness for the micanorm-based logics, using Jenei–Montagna-style construction for proving standard completeness of MTL and related logics (see [8], [15]). However, a similar proof seems to fail with associativity for UL (see Theorem 7 (iv) above). Thus, the

References (26)

  • J. Czelakowski

    Protoalgebraic Logics

    (2001)
  • F. Esteva et al.

    On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic

    Stud. Log.

    (2002)
  • N. Galatos

    Non-associative residuated lattices

  • Cited by (14)

    • Ranking fuzzy numbers using additive priority degrees

      2023, Expert Systems with Applications
    • Involutive basic substructural core fuzzy logics: Involutive mianorm-based logics

      2017, Fuzzy Sets and Systems
      Citation Excerpt :

      Specifically, this paper introduces IMIAL (Involutive mianorm logic), which is intended to cope with the tautologies of left-continuous conjunctive mianorms (binary monotonic identity aggregation operations) and their involutive residua, as InSLℓ, the involutive SLℓ. Note also that the present author considered the similarities and differences between the constructions of Yang and Wang's Jenei–Montagna-style in [22]. However, the similarities and differences between the constructions for involutive logics such as IMICAL and CnIUL have not yet been investigated.

    • Basic substructural core fuzzy logics and their extensions: Mianorm-based logics

      2016, Fuzzy Sets and Systems
      Citation Excerpt :

      In this section, we provide standard completeness results for L (∈ Ls) using the Jenei–Montagna–style construction in [11,23]. See Proposition 2 in [37] for the proof of (I) and the identity of (II). We first prove that the operator ∘ satisfies monotonicity in (II).

    • (Involutive) Basic Substructural Fuzzy Logics and Urquhart-style Semantics

      2023, Journal of Multiple-Valued Logic and Soft Computing
    View all citing articles on Scopus

    The author is grateful to P. Cintula, C. Noguera, R. Horčík, and the referees for their helpful comments and suggestions for improvements to this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2013S1A5A8022638).

    1

    Tel.: +82 62 270 3231.

    View full text